Question regarding acceleration in spacetime

In Brian Greene's book "The Fabric of the Cosmos," he says "For an object's trajectory through spacetime to be straight, the object must not only move in a straight line through space, but its motion must be uniform through time; that is, both its speed and direction must be unchanging and hence it must be moving with constant velocity."

This makes perfect sense but there's something that I don't get. Lets say I'm accelerating but moving in a perfectly straight line through space, wouldn't that also mean my trajectory through spacetime would be straight as well? Ofcourse, I understand that I am not moving uniformly through time and therefore if you were to plot my motion on a graph of space and time, it would obviously be curved, implying acceleration. But as far as my trajectory through spacetime goes, I can't see it being any other way than straight. How can it be a curved trajectory if I am moving perfectly straight through space?

This makes perfect sense but there's something that I don't get. Lets say I'm accelerating but moving in a perfectly straight line through space, wouldn't that also mean my trajectory through spacetime would be straight as well? Ofcourse, I understand that I am not moving uniformly through time and therefore if you were to plot my motion on a graph of space and time, it would obviously be curved, implying acceleration. But as far as my trajectory through spacetime goes, I can't see it being any other way than straight. How can it be a curved trajectory if I am moving perfectly straight through space?

I don't know quite what is meant by "moving uniformly through time", but even in Newtonian physics where there is no time dilation, accelerating in a straight line through space will still lead to a curved path when you plot it on a position vs. time graph. Try plotting x(t) = t^2 (accelerating along the x-axis) on a graph with an x axis and a t axis and you'll see what I mean.

I think you're missing the point. In special relativity, spacetime itself is the ultimate arbiter of accelerated motion. Spacetime itself provides the backdrop to which something can be said to accelerate, even in an empty universe. Therefore, an object's trajectory through spacetime is what tells us wether or not it is accelerating. If it is, its trajectory will be straight, and if not, it will be curved. But my question is if I am accelerating in a straight line through space, how can this trajectory be curved in spacetime. Remember, I am not talking about plotting points on a graph, I understand that argument. I am talking about if you were to look at spacetime as a whole and observe my accelerated motion in a straight line through it, how could this trajectory be curved? Hope I'm making myself clear, as this is hard to describe without pictures.

a straight line through space, how can this trajectory be curved in spacetime

Space and space-time are not the same. Can you not imagine at piece of wire such that, when you look at it from a particular angle (ie. in 2D), it seems straight however if you turn it in 3D you'll realise it is actually curved? In GR what matters is whether the path is curved in 4D space-time.

Yes, I realize this. But if you were able to view the whole of spacetime from an outside perspective (similar to the block universe pics), and in that spacetime block you saw a baseball being thrown. Wouldn't you observe its trajectory in spacetime to be arched, just like that of its actual trajectory in space. Therefore, if I am accelerating in a straight line in space, shouldn't my trajectory in spacetime be straight as well? The whole point of the question is that it can't be straight because if it were, it would imply that I'm not accelerating, as Brian Greene says. But I obviously am accelerating so my trajectory through spacetime must be curved. Yet as I said, if you saw an object's curved trajectory inside the spacetime block from an outside perspective, it would mean that the object is changing speed AND direction. But I am only changing speed. Does this make sense?

But my question is if I am accelerating in a straight line through space, how can this trajectory be curved in spacetime.

I don't understand--why wouldn't it be?

Bos said:

Remember, I am not talking about plotting points on a graph, I understand that argument. I am talking about if you were to look at spacetime as a whole

"Spacetime as a whole" looks exactly like a graph of position vs. time in SR (as measured in an inertial frame). What else would it look like?

Bos said:

and observe my accelerated motion in a straight line through it

A straight line through what, space or spacetime? You can't have accelerated motion that makes a straight line through spacetime, although of course you can accelerate in a straight line through space.

Right. I think my confusion was that I thought an objects' spacetime trajectory was analogous to its space trajectory, only also moving forward through time. I guess I couldn't separate the two as independent of each other. thanx

In Brian Greene's book "The Fabric of the Cosmos," he says "For an object's trajectory through spacetime to be straight, the object must not only move in a straight line through space, but its motion must be uniform through time; that is, both its speed and direction must be unchanging and hence it must be moving with constant velocity."

This makes perfect sense but there's something that I don't get. Lets say I'm accelerating but moving in a perfectly straight line through space, wouldn't that also mean my trajectory through spacetime would be straight as well? Ofcourse, I understand that I am not moving uniformly through time and therefore if you were to plot my motion on a graph of space and time, it would obviously be curved, implying acceleration. But as far as my trajectory through spacetime goes, I can't see it being any other way than straight. How can it be a curved trajectory if I am moving perfectly straight through space?

Clearly I must be missing something. Please help.

Bos,

Here's one way of looking at it ...

When one is stationary in flat spacetime, he is inertial. He travels a straight path thru spacetime, or Minkowski’s 4-space if you prefer. Inertial observers do not believe they are moving, but they are actually traveling thru time. However, since space & time are fused into a continuum, we all travel thru spacetime whether stationary or not. Also, let us assume that all observers travel thru spacetime at identical speed (this is key point here), similarly as light travels at invariant c...

Consider 2 inertial observers, A & B. Both are traveling straight and parallel paths thru spacetime, and hence their relative seperation doesn't change. Time ticks at the same rate for both. Now then, observer B accelerates off in speed alone to a new state of inertial motion. Per A, B’s path thru the spacetime cannot remain straight. To create seperation over time (ie v>0), B has to turn in direction thru 4-space (their paths become unparallel) since B cannot change his invariant speed thru 4-space. Per A who is still going straight, B must then travel a curved path thru the flat 4-space as B accelerates.